Bounds on the domination number of a digraph and its reverse Michitaka Furuya∗ College of Liberal Arts and Science, Kitasato University, 1-15-1 Kitasato, Minami-ku, Sagamihara, Kanagawa 252-0373, Japan

Abstract Let D be a digraph. A dominating set of D is the subset S of V (D) such that each vertex in V (D) − S is an out-neighbor of a vertex in S. The minimum cardinality of a dominating set of G is denoted by γ(D). We let D− denote the reverse of D. In [Discrete Math. 197/198 (1999) 179–183], Chartrand, Harary and Yue proved that every connected digraph D of order n ≥ 2 satisfies γ(D) + γ(D− ) ≤ 4n 3

and characterized the digraphs D attaining the equality. In this paper, we

pose a reduction of the determining problem for γ(D) + γ(D− ) using the total domination concept. As a corollary of such a reduction and known results, we give new bounds for γ(D) + γ(D− ) and an alternative proof of ChartrandHarary-Yue theorem.

Key words and phrases. domination number, digraph, total domination number AMS 2010 Mathematics Subject Classification. 05C69, 05C20.

1

Introduction

All graphs and digraphs considered in this paper are finite and simple. In particular, no digraph has two arcs with same initial vertex and same terminal vertex (but a digraph may contain a directed cycle of order 2). Let G be a graph or a digraph. Let V (G) denote the vertex set of G. If G is a graph, let E(G) denote the edge set of G; if G is a digraph, let A(G) denote the arc set of G.

∗

[email protected]

1

Let G be a graph. For x ∈ V (G), let NG (x) and dG (x) denote the neighborhood and the degree of x, respectively; thus NG (x) = {y ∈ V (G) : xy ∈ E(G)} and dG (x) = |NG (x)|. Let δ(G) denote the minimum degree of G. For n ≥ 3, let Pn and Cn denote the path and the cycle of order n, respectively. + − − Let D be a digraph. For x ∈ V (D), let ND (x), ND (x), d+ D (x) and dD (x) denote

the out-neighborhood, the in-neighborhood, the out-degree and in-degree of x, respec+ − tively; thus ND (x) = {y ∈ V (D) : (x, y) ∈ A(D)}, ND (x) = {y ∈ V (D) : (y, x) ∈ + − − + + A(D)}, d+ D (x) = |ND (x)| and dD (x) = |ND (x)|. Set δ (D) = min{dD (x) : x ∈ ± + − V (D)}, δ − (D) = min{d− D (x) : x ∈ V (D)} and δ (D) = min{δ (D), δ (D)}. Let

D− denote the reverse of D; thus D− is the digraph on V (D) such that A(D− ) = {(x, y) : (y, x) ∈ A(D)}. A digraph D is connected if the graph obtained from D − → − → by replacing any arcs by edges is connected. For n ≥ 3, let Pn and Cn denote the − → directed path and the directed cycle of order n, respectively; thus Pn is the digraph − → − → with V (Pn ) = {u1 , u2 , . . . , un } and E(Pn ) = {(ui , ui+1 ) : 1 ≤ i ≤ n − 1}, and − → − → Cn = Pn + (un , u1 ). Let G be a graph or a digraph. A set S ⊆ V (G) is a dominating set of G if ∪ ∪ ( x∈S NG (x)) ∪ S = V (G) or ( x∈S NG+ (x)) ∪ S = V (G) according as G is a graph or a digraph. The minimum cardinality of a dominating set of G, denoted by γ(G), is called the domination number of G. The domination number is a classical invariant in graph theory, and it has been widely studied (see the books [8, 9] and, for example, [6, 14, 15, 16] for the domination in digraphs). In particular, the domination number of digraphs can be applied to the solution for various problems: answering skyline query, routing in networks, the choice problem of hotels, etc. (see [19]). Let again G be a graph or a digraph of order n. Since V (G) is a dominating set of G, the inequality γ(G) ≤ n trivially holds. The inequality for graphs can be dramatically improved if G is connected: every connected graph G of order n ≥ 2 satisfies γ(G) ≤

n 2

(see [18]). Here we consider a similar problem for digraphs (i.e.,

the estimation problem for the domination number of connected digraphs). Since a connected digraph D of order at least two has an arc (x, y) ∈ A(D), the set V (D) − {y} is a dominating set of D. Thus the following proposition holds. Proposition 1.1 Let D be a connected digraph of order n ≥ 2. Then γ(D) ≤ n−1. The gap between the trivial inequality γ(D) ≤ n and the inequality in Proposition 1.1 is very small, but Proposition 1.1 is best possible. For n ≥ 2, let Dn be the digraph with V (Dn ) = {xi , y : 1 ≤ i ≤ n − 1} and A(Dn ) = {(xi , y) : 1 ≤ i ≤ n − 1}. Then Dn is connected and γ(D) = |V (D)| − 1, and hence Proposition 1.1 is best possible. On the other hand, the domination number of the reverse Dn− of Dn is very small (indeed, γ(Dn− ) = 1 clearly holds). Thus we expect that, in general, if 2

− → P3

G

(attachment vertex) − → Figure 1: The digraph P3 and a digraph G belonging to G(H1 )

the domination number of a digraph D is large, then the domination number of its reverse D− tend to be small. Chartrand, Harary and Yue [3] studied the value γ(D) + γ(D− ) for digraphs D from such a motivation. Let H be a set of connected graphs or a set of connected digraphs. For each H ∈ H, we will fix a vertex v ∈ V (H) and call v the attachment vertex of H (for example, see the next paragraph). Let G(H) be the set of connected graphs G or the set of connected digraphs G, according as the elements of H are graphs or digraphs, such that (H1) H1 , . . . , Hm are vertex-disjoint graphs or digraphs, and Hi is a copy of an element of H, and (H2) G is obtained from

∪

1≤i≤m Hi

by adding some edges or some arcs which join

attachment vertices. For G ∈ G(H), since G is connected, the subgraph or the subdigraph of G induced by the attachment vertices is also connected. − → − → − → We let H1 = {P3 } and define the attachment vertex of P3 as the vertex v of P3 − with d+ − → (v) = d− → (v) = 1 (see Figure 1). Chartrand et al. [3] proved the following P3

P3

theorem. Theorem A ([3]) Let D be a connected digraph of order n ≥ 2. Then γ(D) + − → 4n − γ(D− ) ≤ 4n 3 . Furthermore, if γ(D) + γ(D ) = 3 , then D ∈ {C3 } ∪ G(H1 ). Recently, the domination number of the reverse of a digraph has been focused on. For example, Hao and Qian [10] continued the study of γ(D) + γ(D− ) for digraphs without small directed cycles. Furthermore, the difference of γ(D) and γ(D− ) was studied in [7, 17]. In this paper, we suggest an approach to estimate the value γ(D) + γ(D− ) using the total domination concept. In Section 2, we show that the value γ(D) + γ(D− ) is equal to the total domination number of a special bipartite graph. This, together with known results concerning total domination, leads to many upper bounds for γ(D) + γ(D− ). Our main results in this paper are following: 3

• We give an alternative proof of Theorem A in Section 3. • We show that γ(D) + γ(D− ) ≤ δ ± (D)

8|V (D)| 7

for every connected digraph D satisfying

≥ 1 with finite exceptions, and characterize the digraphs with the equality

(Theorem 4.1 in Section 4). • We give upper bounds on γ(D) + γ(D− ) for a digraph with large δ ± (D) (Theorem 5.1 in Section 5).

2

Reduction to a total domination problem in bipartite graphs

Let G be a graph without isolated vertices, and let X ⊆ V (G). A set S ⊆ V (G) is a ∪ total X-dominating set of G if X ⊆ v∈S NG (v). The minimum cardinality of a total X-dominating set of G is denoted by γt (G; X). The integer γt (G) := γt (G; V (G)) is called the total domination number of G. Lemma 2.1 Let G be a bipartite graph with the bipartition (X, Y ), and suppose that G has no isolated vertices. Then γt (G) = γt (G; X) + γt (G; Y ). Proof. Let SX and SY be a total X-dominating set and a total Y -dominating set of G, respectively. Then SX ∪ SY is a total dominating set of G. Thus γt (G) ≤ γt (G; X) + γt (G; Y ). Let S be a total V (G)-dominating set of G. Then S ∩ Y and S ∩ X are a total X-dominating set and a total Y -dominating set of G, respectively. Thus γt (G) ≥ γt (G; X) + γt (G; Y ).

□

For a digraph D, let G(D) be the graph such that V (G(D)) = {x+ , x− : x ∈ V (D)} and E(G(D)) = {x+ x− : x ∈ V (D)} ∪ {x+ y − : (x, y) ∈ A(D)}, + − and set XD = {x+ : x ∈ V (D)} and XD = {x− : x ∈ V (D)}. Then G(D) is + − a bipartite graph with the ordered bipartition XD := (XD , XD ) and δ(G(D)) =

δ ± (D) + 1. In particular, G(D) has no isolated vertices. Furthermore, G(D) is connected if and only if D is connected. Let MD := {x+ x− : x ∈ V (D)}. Note that MD is a perfect matching of G(D). − Lemma 2.2 Let D be a digraph. Then γ(D) = γt (G(D); XD ).

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− -dominating set of G(D), and set S0 = {x ∈ V (D) : Proof. Let S be a total XD

x+ ∈ S}. Note that |S0 | ≤ |S|. Fix a vertex y ∈ V (D) − S0 . Since y + ∈ / S, there exists a vertex x+ ∈ S such that x+ ̸= y + and x+ y − ∈ E(G(D)). In particular, there exists a vertex x ∈ S0 such that (x, y) ∈ A(D). Since y is arbitrary, S0 is a − dominating set of D. Consequently, γ(D) ≤ γt (G(D); XD ).

Let S ′ be a dominating set of D, and set S0′ = {x+ : x ∈ S ′ }. Fix a vertex − y − ∈ XD . Then there exists a vertex x ∈ S ′ such that either x = y or (x, y) ∈ A(D).

In either case, we have x+ ∈ S0′ and x+ y − ∈ E(G(D)). Since y − is arbitrary, S0′ is − − a total XD -dominating set of G. Consequently, γ(D) ≥ γt (G(D); XD ).

□

By similar argument in the proof of Lemma 2.2, we also obtain the following lemma. + Lemma 2.3 Let D be a digraph. Then γ(D− ) = γt (G(D); XD ).

By Lemmas 2.1–2.3, the following theorem holds. Theorem 2.4 Let D be a digraph. Then γ(D) + γ(D− ) = γt (G(D)). Let G be a bipartite graph with an ordered bipartition X = (X1 , X2 ), and suppose that G has a perfect matching M = {x11 x12 , . . . , xn1 xn2 } where Xi = {x1i , . . . , xni }. Let D(G, X, M ) be the digraph such that V (D(G, X, M )) = {x1 , . . . , xn } and A(D(G, X, M )) = {(xi , xj ) : xi1 xj2 ∈ E(G), i ̸= j}. By the definition of D(G, X, M ), we obtain the following observation. Observation 2.5

(i) A digraph D is isomorphic to D(G(D), XD , MD ).

(ii) A bipartite graph G with an ordered bipartition X having a perfect matching M is isomorphic to G(D(G, X, M )).

3

An alternative proof of Theorem A

Let H1′ = {P3 }, and define the attachment vertex of P3 as a leaf of P3 . Then the following theorem holds. Theorem B ([2, 4]) Let G be a connected graph of order n ≥ 3. Then γt (G) ≤ Furthermore, if γt (G) =

2n 3 ,

then G ∈ {C3 , C6 } ∪ G(H1′ ).

Now we prove Theorem A by Theorem B and some results in Section 2. 5

2n 3 .

−

G ∈ G(H1′ ) − +

+

+

−

−

−

+

D(G, X, M )

+ −

+

Figure 2: Perfect matching M (bold lines) of a graph G in G(H1′ ) H1,1

H1,2

H2

Figure 3: Digraphs H1,1 , H1,2 and H2

Proof of Theorem A.

Let D and n be as in Theorem A. Since G(D) is connected

and |V (G(D))| = 2n ≥ 4, it follows from Theorems 2.4 and B that γ(D) + γ(D− ) = γt (G(D)) ≤ Assume that γ(D) + γ(D− ) =

4n 3 .

4n 2|V (G(D))| = . 3 3

(3.1) 2|V (G(D))| . 3 ′ {C6 } ∪ G(H1 ).

Then (3.1) forces γt (G(D)) =

Since G(D) is bipartite, it follows from Theorem B that G(D) ∈

For any ordered bipartition X of C6 and any perfect matching M of C6 , we have − → D(C6 , X, M ) ≃ C3 ; for any bipartite graph G ∈ G(H1′ ), any ordered bipartition X of G and any perfect matching M of G, we can verify that D(G, X, M ) ∈ G(H1 ) (see Figure 2). This together with Observation 2.5(i) implies D ≃ D(G(D), XD , MD ) ∈ − → {C3 } ∪ G(H1 ). □

4

Digraphs D with δ ± (D) ≥ 1

Let H1,1 , H1,2 and H2 be the digraphs depicted in Figure 3. Let H2 = {W1 , W2 , W3 , W4 } where Wi is the digraph depicted in Figure 4. We define the attachment vertex of Wi as the vertex of Wi enclosed with a circle. In this section, we show the following theorem.

6

W1

W2

W3

W4

Figure 4: Digraphs Wi with the attachment vertex H1′

H2′

Figure 5: Graphs Hi′

Theorem 4.1 Let D be a connected digraph of order n with δ ± (D) ≥ 1. Then − → − → either D ∈ {C3 , C5 , H1,1 , H1,2 , H2 } or γ(D) + γ(D− ) ≤ 8n 7 . Furthermore, if n ≥ 8 and γ(D) + γ(D− ) =

8n 7 ,

then D ∈ G(H2 ).

Let H1′ and H2′ be the graphs depicted in Figure 5. Let H2′ = {W1′ , W2′ } where Wi′ is the graph depicted in Figure 6. We define the attachment vertex of Wi′ as the vertex of Wi′ enclosed with a circle. Henning [11] proved the following theorem. Theorem C ([11]) Let G be a connected graph of order n with δ(G) ≥ 2. Then either G ∈ {C3 , C5 , C6 , C10 , H1′ , H2′ } or γt (G) ≤ γt (G) =

4n 7 ,

then G ∈

Proof of Theorem 4.1.

4n 7 .

Furthermore, if n ≥ 15 and

G(H2′ ). Let D and n be as in Theorem 4.1. Since G(D) is a connected W1′

W2′

Figure 6: Graphs Wi′ with the attachment vertex

7

D(H1′ , X, M ) ≃ H1,1 H1′ D(H1′ , X, M ) ≃ H1,2

D(H2′ , X, M ) ≃ H2

H2′

Figure 7: Perfect matchings M (bold lines) of graphs H1′ and H2′

bipartite graph with δ(G(D)) ≥ 2 and |V (G(D))| = 2n, it follows from Theorems 2.4 and C that either G(D) ∈ {C6 , C10 , H1′ , H2′ }

(4.1)

or γ(D) + γ(D− ) = γt (G(D)) ≤

4|V (G(D))| 8n = . 7 7

(4.2)

If (4.2) holds, then the first statement of the theorem holds. Thus, for the moment, we may assume that (4.1) holds. Let G ∈ {C6 , C10 , H1′ , H2′ }, and let X be an ordered bipartition of G and M be a perfect matching of G. Then we can check the following: − → − → If G = C6 , then D(G, X, M ) = C3 ; if G = C10 , then D(G, X, M ) = C5 ; if G = H1′ , then D(G, X, M ) ∈ {H1,1 , H1,2 }; if G = H2′ , then D(G, X, M ) = H2 (see Figure 7). In − → − → either case, D(G, X, M ) ∈ {C3 , C5 , H1,1 , H1,2 , H2 }. Hence D ≃ D(G(D), XD , MD ) ∈ − → − → {C3 , C5 , H1,1 , H1,2 , H2 } by Observation 2.5(i). This completes the proof of the first statement of the theorem. Assume that n ≥ 8 (i.e., |V (G(D))| = 2n ≥ 16) and γ(D) + γ(D− ) = (4.2) forces γt (G(D)) =

4|V (G(D))| . 7

8n 7 .

Then

It follows from Theorem C that G(D) ∈ G(H2′ ). 8

G ∈ G(H2′ )

+

−

+

−

D(G, X, M )

Figure 8: Perfect matching M (bold lines) of a graph G in G(H2′ )

For any bipartite graph G ∈ G(H2′ ), any ordered bipartition X of G and any perfect matching M of G, we can verify that D(G, X, M ) ∈ G(H2 ) (see Figure 8). This together with Observation 2.5(i) implies D ≃ D(G(D), XD , MD ) ∈ G(H2 ). This completes the proof of Theorem 4.1.

5

□

Digraphs D with large δ ± (D)

There are many results concerning the total domination number of graphs with large minimum degree, as follows. Theorem D ([1]) Let G be a connected graph of order n with δ(G) ≥ 3. Then γt (G) ≤ n2 . Theorem E ([20]) Let G be a connected graph of order n with δ(G) ≥ 4. Then γt (G) ≤

3n 7 .

Theorem F ([5]) Let G be a connected graph of order n with δ(G) ≥ 5. Then γt (G) ≤

17n 44 .

Theorem G ([12]) Let d ≥ 2 be an integer, and let G be a connected graph of order n with δ(G) ≥ d. Then γt (G) ≤

(1+ln d)n . d

By Theorem 2.4 and above results, we obtain the following theorem. 9

Theorem 5.1 Let d ≥ 2 be an integer, and let D be a connected digraph of order n with δ ± (D) ≥ d. Then

γ(D) + γ(D− ) ≤

  n       6n

(d = 2) (d = 3)

7

 17n     22    2(1+ln(d+1))n d+1

(d = 4) (d ≥ 5).

Henning and Yeo [13] characterized the set H∗ of the connected graphs G of order n with δ(G) ≥ 3 and γt (G) = n2 . The set H∗ contains infinitely many bipartite graphs having perfect matchings. In particular, for any bipartite graph G ∈ H∗ with an ordered bipartition X having a perfect matching M , it follows from Observation 2.5(ii) that D(G, X, M ) is a connected digraph with δ ± (D(G, X, M )) ≥ 2 and γ(D(G, X, M )) + γ(D(G, X, M )− ) = γt (G(D(G, X, M ))) = γt (G) |V (G)| 2 = |V (D(G, X, M ))|. =

Hence Theorem 5.1 for the case d = 2 is best possible. Furthermore, by the similar strategy in the proof of Theorems A and 4.1, we can characterize the connected digraphs D of order n with δ ± (D) ≥ 2 and γ(D) + γ(D− ) = n. However, since the bipartite graphs in H∗ have many perfect matchings, it seems that the characterization of such digraphs D is not easy. We leave the characterization problem as an exercise for the readers.

Acknowledgment This work was supported by JSPS KAKENHI Grant number 26800086.

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